ON THE USE OF REALISTIC FERMI PROBLEMS FOR INTRODUCING MATHEMATICAL MODELLING IN SCHOOL

In this paper uses an analytical tool refereed to as the MAD (Modelling Activity Diagram) framework adapted from Schoenfeld's parsing protocol coding scheme to address the issues of how to introduce mathematical modelling to upper secondary students. The work of three groups of students engaged in solving so called realistic Fermi problems were analysed using this framework, and it was observed that the processes involved in a typical mathematical modelling cycle were richly represented in the groups' solving processes. The importance of the social interactions within the groups was noted, as well as the extensive use of extra- mathematical knowledge used by the students during the problem solving session.

[1]  Douglas R. M. Edge,et al.  Problem Solving, Enrico Fermi and the Bull Moose , 1983 .

[2]  A. Schoenfeld.  Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics (Reprint) , 2009 .

[3]  Mark N. Hoover,et al.  Principles for Developing Thought-Revealing Activities for Students and Teachers , 2000 .

[4]  M. Goos,et al.  Do it this way! Metacognitive strategies in collaborative mathematical problem solving , 1996 .

[5]  M. Goldberger.  Enrico Fermi (1901–1954): The Complete Physicist , 1999 .

[6]  Gabriele Kaiser,et al.  A global survey of international perspectives on modelling in mathematics education , 2006 .

[7]  Peter Galbraith,et al.  Mathematical modelling (ICTMA 12) : education, engineering and economics , 2007 .

[8]  John Berry,et al.  Developing mathematical modelling skills: The role of CAS , 2002 .

[9]  Rita Borromeo Ferri,et al.  Theoretical and empirical differentiations of phases in the modelling process , 2006 .

[10]  Werner Blum,et al.  How do Students and Teachers Deal with Modelling Problems , 2007 .

[11]  Alan H. Schoenfeld,et al.  Mathematical Problem Solving , 1985 .

[12]  Werner Blum,et al.  Modelling and Applications in Mathematics Education : the 14th ICMI Study , 2007 .

[13]  Mogens Allan Niss,et al.  Aims and scope of appications and modelling in mathematics curricula , 1987 .

[14]  R. Lesh,et al.  Handbook of Research Design in Mathematics and Science Education. , 2000 .

[15]  J. Smithuis.  Upper secondary school , 2008 .

[16]  Ingela Eriksson,et al.  En tolkning av målen med den svenska gymnasiematematiken och tolkningens konsekvenser för uppgiftskonstruktion , 2004 .

[17]  Peter Galbraith,et al.  Modelling and Applications in Mathematics Education , 2007 .

[18]  Eva Jablonka,et al.  Meta-Analyse von Zugängen zur mathematischen Modellbildung und Konsequenzen für den Unterricbt , 1997 .

[19]  Jerry Lege.  Approaching minimal conditions for the introduction of mathematical modeling , 2005 .

[20]  K. A. Ericsson,et al.  Verbal reports as data. , 1980 .

[21]  Morten Blomhøj,et al.  What’s all the Fuss about Competencies? , 2007 .

[22]  Morten Blomhøj,et al.  REPORT FROM THE WORKING GROUP MODELLING AND APPLICATIONS - DIFFERENTIATING PERSPECTIVES AND DELINEATING COMMONALTIES , 2007 .

[23]  Katja Maaß,et al.  What are modelling competencies , 2006 .

[24]  Andrea Peter-Koop,et al.  Teaching and Understanding Mathematical Modelling through Fermi-Problems , 2009 .

[25]  Thomas Lingefjärd,et al.  Faces of mathematical modeling , 2006 .

[26]  Rita Borromeo Ferri,et al.  Modelling Problems from a Cognitive Perspective , 2007 .

[27]  Thomas P. Hogan,et al.  Quantitative Estimation: One, Two, or Three Abilities? , 2003 .

[28]  Richard Lesh,et al.  Modeling conceptions revisited , 2006 .

[29]  Timothy D. Wilson,et al.  Telling more than we can know: Verbal reports on mental processes. , 1977 .

[30]  Morten Blomhøj,et al.  Towards a didactical theory for mathematical modelling , 2006 .

[31]  Jill P Brown.  An insight into student understanding of functions in a graphing calculator environment , 2003 .

[32]  Bharath Sriraman,et al.  The Mathematics of Estimation: Possibilities for Interdisciplinary Pedagogy and Social Consciousness , 2009 .

[33]  Per Linell,et al.  Approaching Dialogue: Talk, Interaction and Contexts in Dialogical Perspectives , 1998 .

[34]  Christopher Haines,et al.  Remarks on a modelling cycle and interpretation of behaviours , 2007 .

[35]  C. J. Efthimiou,et al.  Cinema, Fermi problems and general education , 2006, physics/0608058.

[36]  Gabriele Kaiser,et al.  Theory usage and theoretical trends in Europe: a survey and preliminary analysis of CERME4 research reports , 2006 .

[37]  Werner Blum,et al.  Applied mathematical problem solving, modelling, applications, and links to other subjects — State, trends and issues in mathematics instruction , 1991 .

[38]  Werner Blum.  Applications and modelling in mathematics teaching , 1991 .

[39]  A W Robinson,et al.  Don’t just stand there—teach Fermi problems! , 2008 .

[40]  A. O. Moscardini,et al.  Mathematical Modelling Courses , 1988 .

[41]  J. Carlson,et al.  Fermi problems on gasoline consumption , 1997 .

[42]  G. Davison,et al.  Think-aloud approaches to cognitive assessment and the articulated thoughts in simulated situations paradigm. , 1997, Journal of consulting and clinical psychology.

[43]  Judith T. Sowder,et al.  Estimation and number sense. , 1992 .

[44]  Hugh Burkhardt,et al.  Modelling in Mathematics Classrooms: reflections on past developments and the future , 2006 .

[45]  Richard Lesh,et al.  Beyond Constructivism: Models and Modeling Perspectives on Mathematics Problem Solving, Learning, and Teaching , 2003 .